Harmonics
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Harmonics. October 29, 2012. Where Were We?. We’re halfway through grading the mid-terms. For the next two weeks: more acoustics It’s going to get worse before it gets better Note: I’ve posted a supplementary reading on today’s lecture to the course web page. What have we learned?

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Harmonics

Harmonics

October 29, 2012


Where were we

Where Were We?

  • We’re halfway through grading the mid-terms.

  • For the next two weeks: more acoustics

    • It’s going to get worse before it gets better

  • Note: I’ve posted a supplementary reading on today’s lecture to the course web page.

  • What have we learned?

    • How to calculate the frequency of a periodic wave

    • How to calculate the amplitude, RMS amplitude, and intensity of a complex wave

  • Today: we’ll start putting frequency and intensity together


Complex waves

Complex Waves

  • When more than one sinewave gets combined, they form a complex wave.

  • At any given time, each wave will have some amplitude value.

    • A1(t1) := Amplitude value of sinewave 1 at time 1

    • A2(t1) := Amplitude value of sinewave 2 at time 1

  • The amplitude value of the complex wave is the sum of these values.

    • Ac(t1) = A1 (t1) + A2 (t1)


Complex wave example

Complex Wave Example

  • Take waveform 1:

    • high amplitude

    • low frequency

+

  • Add waveform 2:

    • low amplitude

    • high frequency

=

  • The sum is this complex waveform:


Greatest common denominator

Greatest Common Denominator

  • Combining sinewaves results in a complex periodic wave.

  • This complex wave has a frequency which is the greatest common denominator of the frequencies of the component waves.

  • Greatest common denominator = biggest number by which you can divide both frequencies and still come up with a whole number (integer).

  • Example:

    • Component Wave 1: 300 Hz

    • Component Wave 2: 500 Hz

    • Fundamental Frequency: 100 Hz


Harmonics

Why?

  • Think: smallest common multiple of the periods of the component waves.

  • Both component waves are periodic

    • i.e., they repeat themselves in time.

  • The pattern formed by combining these component waves...

    • will only start repeating itself when both waves start repeating themselves at the same time.

  • Example: 3 Hz sinewave + 5 Hz sinewave


For example

For Example

  • Starting from 0 seconds:

    • A 3 Hz wave will repeat itself at .33 seconds, .66 seconds, 1 second, etc.

    • A 5 Hz wave will repeat itself at .2 seconds, .4 seconds, .6 seconds, .8 seconds, 1 second, etc.

  • Again: the pattern formed by combining these component waves...

    • will only start repeating itself when they both start repeating themselves at the same time.

    • i.e., at 1 second


Harmonics

3 Hz

.33 sec.

.66

1.00

5 Hz

.20

1.00

.40

.60

.80


Harmonics

1.00

Combination of 3 and 5 Hz waves

(period = 1 second)

(frequency = 1 Hz)


Tidbits

Tidbits

  • Important point:

    • Each component wave will complete a whole number of periods within each period of the complex wave.

  • Comprehension question:

    • If we combine a 6 Hz wave with an 8 Hz wave...

    • What should the frequency of the resulting complex wave be?

  • To ask it another way:

    • What would the period of the complex wave be?


Harmonics

6 Hz

.17 sec.

.33

.50

8 Hz

.125

.25

.375

.50


Harmonics

.50 sec.

1.00

Combination of 6 and 8 Hz waves

(period = .5 seconds)

(frequency = 2 Hz)


Fourier s theorem

Fourier’s Theorem

  • Joseph Fourier (1768-1830)

    • French mathematician

    • Studied heat and periodic motion

  • His idea:

    • any complex periodic wave can be constructed out of a combination of different sinewaves.

  • The sinusoidal (sinewave) components of a complex periodic wave = harmonics


The dark side

The Dark Side

  • Fourier’s theorem implies:

    • sound may be split up into component frequencies...

    • just like a prism splits light up into its component frequencies


Spectra

Spectra

  • One way to represent complex waves is with waveforms:

    • y-axis: air pressure

    • x-axis: time

  • Another way to represent a complex wave is with a power spectrum (or spectrum, for short).

  • Remember, each sinewave has two parameters:

    • amplitude

    • frequency

  • A power spectrum shows:

    • intensity (based on amplitude) on the y-axis

    • frequency on the x-axis


Two perspectives

Two Perspectives

WaveformPower Spectrum

+

+

=

=

harmonics


Example

Example

  • Go to Praat

  • Generate a complex wave with 300 Hz and 500 Hz components.

  • Look at waveform and spectral views.

  • And so on and so forth.


Fourier s theorem part 2

Fourier’s Theorem, part 2

  • The component sinusoids (harmonics) of any complex periodic wave:

    • all have a frequency that is an integer multiple of the fundamental frequency of the complex wave.

  • This is equivalent to saying:

    • all component waves complete an integer number of periods within each period of the complex wave.


Example1

Example

  • Take a complex wave with a fundamental frequency of 100 Hz.

  • Harmonic 1 = 100 Hz

  • Harmonic 2 = 200 Hz

  • Harmonic 3 = 300 Hz

  • Harmonic 4 = 400 Hz

  • Harmonic 5 = 500 Hz

  • etc.


Deep thought time

Deep Thought Time

  • What are the harmonics of a complex wave with a fundamental frequency of 440 Hz?

  • Harmonic 1: 440 Hz

  • Harmonic 2: 880 Hz

  • Harmonic 3: 1320 Hz

  • Harmonic 4: 1760 Hz

  • Harmonic 5: 2200 Hz

  • etc.

  • For complex waves, the frequencies of the harmonics will always depend on the fundamental frequency of the wave.


A new spectrum

A new spectrum

  • Sawtooth wave at 440 Hz

  • Also check out a sawtooth wave at 150 Hz


Another representation

Another Representation

  • Spectrograms

    • = a three-dimensional view of sound

  • Incorporate the same dimensions that are found in spectra:

    • Intensity (on the z-axis)

    • Frequency (on the y-axis)

  • And add another:

    • Time (on the x-axis)

  • Back to Praat, to generate a complex tone with component frequencies of 300 Hz, 2100 Hz, and 3300 Hz


Something like speech

Something Like Speech

  • Check this out:

  • One of the characteristic features of speech sounds is that they exhibit spectral change over time.

source: http://www.haskins.yale.edu/featured/sws/swssentences/sentences.html


Harmonics and speech

Harmonics and Speech

  • Remember: trilling of the vocal folds creates a complex wave, with a fundamental frequency.

  • This complex wave consists of a series of harmonics.

  • The spacing between the harmonics--in frequency--depends on the rate at which the vocal folds are vibrating (F0).

  • We can’t change the frequencies of the harmonics independently of each other.

  • Q: How do we get the spectral changes we need for speech?


Resonance

Resonance

  • Answer: we change the intensity of the harmonics

  • Listen to this:

  • source: http://www.let.uu.nl/~audiufon/data/e_boventoon.html


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